A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2D

A direct reconstruction algorithm for complex conductivities in W-2,W-infinity(Omega), where Omega is a bounded, simply connected Lipschitz domain in R-2, is presented. The framework is based on the uniqueness proof by Francini (2000 Inverse Problems 6 107-19), but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.

GROUND STATE AND NON-GROUND STATE SOLUTIONS OF SOME STRONGLY COUPLED ELLIPTIC SYSTEMS

We study an elliptic system of the form Lu = vertical bar v vertical bar(p-1) v and Lv = vertical bar u vertical bar(q-1) u in Omega with homogeneous Dirichlet boundary condition, where Lu := -Delta u in the case of a bounded domain and Lu := -Delta u + u in the cases of an exterior domain or the whole space R-N. We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

Advanced QSAR Studies on PPAR delta Ligands Related to Metabolic Diseases

PPAR delta is a nuclear receptor that, when activated, regulates the metabolism of carbohydrates and lipids and is related to metabolic syndrome and type 2 diabetes. To understand the main interactions between ligands and PPAR delta, we have constructed 2D and 3D QSAR models and compared them with HOMO, LUMO and electrostatic potential maps of the compounds studied, as well as docking results. All QSAR models showed good statistical parameters and prediction outcomes. The QSAR models were used to predict the biological activity of an external test set, and the predicted values are in good agreement with the experimental results. Furthermore, we employed all maps to evaluate the possible interactions between the ligands and PPAR delta. These predictive QSAR models, along with the HOMO, LUMO and MEP maps, can provide insights into the structural and chemical properties that are needed in the design of new PPAR delta ligands that have improved biological activity and can be employed to treat metabolic diseases.

Descrição morfológica do diafragma do sagui-de-tufo-branco (Callithrix jacchus)

O músculo diafragma, encontrado apenas nos mamíferos, é o principal músculo no processo respiratório, servindo de fronteira entre as cavidades torácica e abdominal. Sua importância também ganha destaque em pesquisas realizadas no âmbito dos enxertos, empregando-se diversos tipos de membranas biológicas para o reparo de defeitos diafragmáticos, os quais podem gerar hérnias diafragmáticas. Apesar de muitos estudos já conduzidos para com os primatas não humanos, especialmente no que tange a espécie do novo mundo Callithrix jacchus (Sagui-de-tufo-branco), oriundo do nordeste brasileiro, as pesquisas envolvendo o uso do diafragma em tal espécie é inexistente. Deste modo objetivou-se caracterizar a morfologia e a biometria do diafragma na espécie Callithrix jacchus de ambos os sexos, analisando possíveis divergências estruturais entre machos e fêmeas. Para tal foram utilizados quatros animais, 2 machos e 2 fêmeas, adultos, que vieram a óbito por causas naturais, provenientes de um criadouro comercial. Após fixação em solução de formaldeído 10% os animais foram devidamente dissecados para fotodocumentação e em seguida o diafragma coletado para efetuação da biometria (comprimento e largura) com o uso de um paquímetro e para o processamento histológico por meio da coloração de hematoxilina-eosina e tricrômio de masson, da porção muscular. As mensurações feitas permitiram concluir que não houve diferenças signifcativas entre machos e femeas. A topografia e a presença de três aberturas (forame da veia cava caudal, hiato aórtico e esofágico) na extensão do diafragma corroboram com descrições na literatura classica para outros mamíferos. A presença de um centro tendíneo em "V" difere do encontrado para animais como o peixe-boi e porquinho-da-india, mas é similar ao encontrado para o gambá-de-orelhas-brancas e rato albino. No que diz respeito aos achados histológicos conclui-se que as fibras musculares estão dispostas de forma organizada, apresentam diâmetro grande e núcleos basais, tendo, portanto, características similares do músculo estriado esquelético tanto nos animais machos como nas fêmeas.

Arquitetura brutalista e estratégia de transportes no triângulo mineiro: estações ferroviárias da Mogiana e Terminal Rodoviário Presidente Castelo Branco

Em 1989, Brandão descrevia o Triângulo Mineiro como “fruto da ambiguidade de seu estigma de fazer parte de Minas, mas ser articulada economicamente a São Paulo.” A mesorregião do Triângulo Mineiro e Alto Paranaíba, faz fronteira com os estados de Goiás, São Paulo e Mato Grosso do Sul, interligando também com a Central Mineira e com o Oeste de Minas, sendo a característica de “rota de passagem” como principal fator do desenvolvimento de sua economia. O posicionamento estratégico da região, como eixo de ligação da capital paulista ao chamado Brasil Central, pode ser considerado um importante fator no estreitamento dos laços entre a região e São Paulo, somado ao sentimento de não pertencimento do Triângulo ao estado de Minas Gerais, o qual resultou por décadas em manifestações separatistas na região. A arquitetura moderna produzida no Triângulo Mineiro e Alto Paranaíba deu um salto significativo no momento de construção da nova capital federal, em finais da década de 1950, onde o papel de mediação, principalmente da cidade de Uberlândia, no processo de infra-estruturação da nova cidade foi determinante nos avanços construtivos do Triângulo. Esse momento coincidiu com o início do processo de verticalização das principais cidades da região e o aumento de arquitetos residentes nas cidades. Por meio, em especial, dos edifícios para as estações ferroviárias da Cia Mogiana, de Oswaldo Arthur Bratke em Uberaba e Uberlândia (déc. 1960) e do Terminal Rodoviário Presidente Castelo Branco em Uberlândia, dos arquitetos Fernando Graça, Flávio Almada e Ivan Curpertino (1970), este trabalho objetiva conduzir uma discussão acerca da produção de arquitetura moderna no Triângulo Mineiro ligada às estratégias de transportes intermunicipais como própria cultura de desenvolvimento econômico da região. Nos interessa valer do debate entre o uso da estética brutalista, e da própria escolha por uma arquitetura moderna, como artifício no plano de desenvolvimento das empresas de transporte, e dos governos locais. Sobretudo, discutir as interlocuções do Triângulo Mineiro com São Paulo, rebatendo-as na formação do conjunto arquitetonico moderno produzido na região. Este trabalho é fruto da pesquisa de mestrado da autora cujo tema central é a difusão da arquitetura moderna no Triângulo Mineiro e Alto Paranaíba, pelo Iau/Usp, e financiado pela Capes.

Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of saddle-node equilibrium points

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddle-node equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Ein inverses Problem für die degeneriert parabolische Richardsgleichung

In der vorliegenden Arbeit werden zwei physikalischeFließexperimente an Vliesstoffen untersucht, die dazu dienensollen, unbekannte hydraulische Parameter des Materials, wiez. B. die Diffusivitäts- oder Leitfähigkeitsfunktion, ausMeßdaten zu identifizieren. Die physikalische undmathematische Modellierung dieser Experimente führt auf einCauchy-Dirichlet-Problem mit freiem Rand für die degeneriertparabolische Richardsgleichung in derSättigungsformulierung, das sogenannte direkte Problem. Ausder Kenntnis des freien Randes dieses Problems soll dernichtlineare Diffusivitätskoeffizient derDifferentialgleichung rekonstruiert werden. Für diesesinverse Problem stellen wir einOutput-Least-Squares-Funktional auf und verwenden zu dessenMinimierung iterative Regularisierungsverfahren wie dasLevenberg-Marquardt-Verfahren und die IRGN-Methode basierendauf einer Parametrisierung des Koeffizientenraumes durchquadratische B-Splines. Für das direkte Problem beweisen wirunter anderem Existenz und Eindeutigkeit der Lösung desCauchy-Dirichlet-Problems sowie die Existenz des freienRandes. Anschließend führen wir formal die Ableitung desfreien Randes nach dem Koeffizienten, die wir für dasnumerische Rekonstruktionsverfahren benötigen, auf einlinear degeneriert parabolisches Randwertproblem zurück.Wir erläutern die numerische Umsetzung und Implementierungunseres Rekonstruktionsverfahrens und stellen abschließendRekonstruktionsergebnisse bezüglich synthetischer Daten vor.

Funzioni zeta e fattorizzazione unica in anelli quadratici di curve su campi finiti

In questa tesi si esaminano alcune questioni riguardanti le curve definite su campi finiti. Nella prima parte si affronta il problema della determinazione del numero di punti per curve regolari. Nella seconda parte si studia il numero di classi di ideali dell’anello delle coordinate di curve piane definite da polinomi assolutamente irriducibili, per ottenere, nel caso delle curve ellittiche, risultati analoghi alla classica formula di Dirichlet per il numero di classi dei campi quadratici e delle congetture di Gauss.

Die Faktorisierungsmethode für die elektrische Impedanztomographie im Halbraum

In der vorliegenden Arbeit wird die Faktorisierungsmethode zur Erkennung von Inhomogenitäten der Leitfähigkeit in der elektrischen Impedanztomographie auf unbeschränkten Gebieten - speziell der Halbebene bzw. dem Halbraum - untersucht. Als Lösungsräume für das direkte Problem, d.h. die Bestimmung des elektrischen Potentials zu vorgegebener Leitfähigkeit und zu vorgegebenem Randstrom, führen wir gewichtete Sobolev-Räume ein. In diesen wird die Existenz von schwachen Lösungen des direkten Problems gezeigt und die Gültigkeit einer Integraldarstellung für die Lösung der Laplace-Gleichung, die man bei homogener Leitfähigkeit erhält, bewiesen. Mittels der Faktorisierungsmethode geben wir eine explizite Charakterisierung von Einschlüssen an, die gegenüber dem Hintergrund eine sprunghaft erhöhte oder erniedrigte Leitfähigkeit haben. Damit ist zugleich für diese Klasse von Leitfähigkeiten die eindeutige Rekonstruierbarkeit der Einschlüsse bei Kenntnis der lokalen Neumann-Dirichlet-Abbildung gezeigt. Die mittels der Faktorisierungsmethode erhaltene Charakterisierung der Einschlüsse haben wir in ein numerisches Verfahren umgesetzt und sowohl im zwei- als auch im dreidimensionalen Fall mit simulierten, teilweise gestörten Daten getestet. Im Gegensatz zu anderen bekannten Rekonstruktionsverfahren benötigt das hier vorgestellte keine Vorabinformation über Anzahl und Form der Einschlüsse und hat als nicht-iteratives Verfahren einen vergleichsweise geringen Rechenaufwand.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Teoria spettrale in spazi di Hilbert

In questa tesi ci si occuperà di presentare alcuni aspetti salienti della teoria spettrale per gli operatori limitati negli spazi di Hilbert. Nel primo capitolo verranno presentate alcune nozioni fondamentali di analisi funzionale, necessarie per lo studio degli operatori. Il secondo capitolo si occupa invece di analizzare la teoria spettrale per operatori compatti. In particolare, verrà presentato il Teorema Spettrale per Operatori Normali Compatti e il Teorema dell'Alternativa di Fredholm. In seguito verrà applicata tale teoria alla risolubilità del problema di Dirichlet. Nel terzo capitolo verrà esteso quanto ottenuto per gli operatori compatti ad operatori limitati autoaggiunti e per gli operatori normali limitati, passando attraverso le famiglie spettrali.